14.25 A mass attached to a spring is free to oscillate, with angular velocity $ω$ , in a horizontal plane without friction or damping. It is pulled to a distance $x_{0}$ and pushed towards the centre with a velocity $v_{0}$ at time t = 0. Determine the amplitude of the resulting oscillations in terms of the parameters $ω, x_{0}$ and $v_{0}$ .

[Hint : Start with the equation x = a cos ( $ω$ t + $θ$ ) and note that the initial velocity is negative.]

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Vishal Baghel | Contributor-Level 10

5 months ago

The displacement equation for an oscillating mass is given by : x = Acos ( $ω t + θ)$ , where

A = the amplitude

x = the displacement

$θ = p h a s e c o n s t a n t$

Velocity, V = $\frac{d x}{d t}$ = -A $ω s i n ? (ω$ t + $θ)$

At t = 0, x = $x_{0}$ , $x_{0} = A \cos ? θ$ = $x_{0}$ ….(i)

And $\frac{d x}{d t}$ = $- v_{0}$ = A $ω \sin ? θ$ …….(ii)

Squaring and adding, we get

$A^{2} ({c o s}^{2} θ + {s i n}^{2} θ) = x_{0}^{2} + (\frac{v_{0}^{2}}{ω^{2}})$ , A = $\sqrt{x_{0}^{2} + ({\frac{v_{0}}{ω})}^{2}}$

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